Complex Doesn't Have to be so Complex!

Objective: SWBAT write ratios as fractions 3 different ways, as a complex fraction, a unit rate, and a unit fraction.

Big Idea:
Students will grapple through multi-step problem to find the amount spent on a pair of boots from Macys using the ratio of money spent to money left over in an account and write the ratio as fractions in four ways.

This task is a rich task. Students will need to be able to understand how to identify the order of a ratio within the text, what a complex fraction is, what it looks like, and how to find the numerator or denominator of the complex fraction depending upon the order of the ratio. Students will need to know what to do with the fraction of the money spent and the amount in her account to find the dollar amount spent. Students will need to calculate the amount left in her account. Students will need to understand that the money deposited by Crystal was added to the current balance and the sum of these two amounts must be used to solve the task. Students will need to know what a unit fraction is, and how to calculate the unit fraction, as well as what a unit rate is, and how to calculate the unit rate.

You may opt to scaffold this task into parts. This may help lower level learners see each part of the task. I use a strategy called “unpacking the problem”. Students will circle power verbs, underline what the power verb is telling them to do, box numbers or amounts that are needed to help solve the task, circle any questions, and create a checklist from the power verbs that were circled. This will help students identify each part of the task that needs to be completed. I have my students check off each item from their checklist as they complete each.

Because this task has so many components, you may find that this lesson will take 1 ½ days to 2 days depending upon the understanding of each component in the task.

: As my students line up at my door to walk in the room, I hand them the task for the day. This problem will be the focus for the entire lesson. The students will have 10 minutes to grapple through the question on their own. This is their individual think time. Better known in my class room asIndividual Think Timetime. Students will practice MP 1, MP 2, MP 4, and MP 6 during this time. Students with strong understanding will be able to find that if 3/5 of the money was spent, 2/5 of the money was not spent. Students with strong understanding will multiply the ratio of the money spent by the balance of her account to find the amount spent. Some students may opt to use this same strategy to find the amount left over, while others may subtract the money spent from the balance to find the amount left over. Strong learners will be able use the definitions of the unit rates and unit fractions to calculate each. For lower level learners you will need to break each of these down one by one. I found that if student phrase 3/5 as 3 out of 5 and model it, they are able to find the ratio of money left over with ease in comparison to phrasing 3/5 as three fifths.

Resources

Once students have had 10 minutes to grapple through the task on their own, have the students pair up in pairs of groups to go talk about what they discovered during their I.T.T. time. This is a critical time for you. This is the time in which you will want to navigate the room and involve yourself in their mathematical discussions checking for understanding. The all famous “Why did you do that?” “How did you do that?” “Show me why you chose those amounts.” These are all good ways to check for student understanding, keep classroom management, and be sure that students are effectively practicing MP 3.

Resources (1)

Resources

During the whole group discussion, choose a few students to share what they discovered solving this task. Ultimately we want to know if they are able to write the ratio of money spent to money left over 4 ways. We are assessing if students are able to take the information in the text to write a complex fraction, a dollar to dollar fraction, a unit rate, and a unit fraction. During this time students should have the opportunity to explain how they started the task, what they did to write each, are they correct, and if not where did they make their mistakes. Students may make the mistake in writing the complex fraction as 3/5 over $85. This will indicate that they know a complex fraction will have a fraction as the numerator, and that they understood how to find the correct amount by adding the current balance to the deposit. However, they needed to find the fractional amount of the money left over. When students make this mistake I validate the amount of $85.00 is the total amount in the account. I guide students to the text. What does the text ask you to find? The students should respond money spent to money left over. I ask, does crystal have $85 left over if she spent 3/5 of it? This will guide student thinking. This is an example of the type of discussion that you will want to have to assess true understanding.

During the closing discussion, this is the time in which the students will be given the correct process and correct answers to the task. It is important for students to know what they understand and what they did not grasp. Encourage students to keep their mistakes on their papers, and write the correct answers and process to finding the answers next to their mistakes. Have students explain what they did wrong and how they corrected it. You may want to give them a fresh task sheet to do this process. It is important for students to have this as a study resource. I have my students paste this in their Interactive Math Journals to build their resource book. Please see the worked task sheet in the resource section.